Why does Subdivide Smooth look different when compared to the Subsurf modifier?
When using subdivision in edit mode, the original vertices stay at the exact same place. The rest of the mesh swells up around it.

When using the modifier, vertices and edges are more like control points of a nurbs curve. The image below illustrates it. Note that concave shapes (the thinner section of the model below) will swell outwards.



It's not possible to make a truly spherical shape when using subsurf, but the more verts you have in the original mesh, the less noticeable the irregularities will be and the more spherical it will look. You can also use a Shinkwrap modifier to "project" your mesh onto a real sphere, or a Cast modifier or the To Sphere (Alt+Shift+S) operator to warp it into shape. I often do this when wanting a mesh that can be subdivided but still have a nice equator.